# Order Of Operations – Definition With Examples

There are many operations in mathematics, such as addition, subtraction, multiplication, and division. They help us evaluate mathematical expressions.

Consider the following expression: 4+ 5 × 32 – 2

The expression consists of many operations. But which part do you calculate first?

You may start from the left and get one answer. But your friend may begin from the right and get a completely different answer!

Note: Both the methods given above are incorrect.

Hence, to avoid confusion, a standard rule was set to perform such calculations. This rule is known as the order of operations.

## What Is the Order of Operations in Math?

If you have an expression where all the operations are the same (example: only addition, only subtraction, only multiplication, or only division) then the correct way to solve it would be from left to right. But for expressions with multiple operations, we need to follow the order of operations.

The order of operations is the rule that tells us the sequence in which we should solve an expression with multiple operations.

A way to remember that order is PEMDAS. Each letter in PEMDAS stands for a mathematical operation.

Order of Operations Steps:

## Parentheses

The first step is to solve the operation within parentheses or brackets. Parentheses are used to group things together. Work out all groupings from inside to out.

## Exponents

Work out the exponential expressions after the parentheses.

## Multiplication and Division

Next, moving from left to right, multiply and/or divide, whichever comes first.

## Addition and Subtraction

Lastly, moving from left to right, add and/or subtract, whichever comes first.

Why Follow the Order of Operations?

We follow the rules of the order of operations to solve expressions so that everyone arrives at the same answer.

Here’s an example of how we can get different answers if the correct order of operations is NOT followed:

## Solved Examples

Example 1: Solve: 2 + 6 × (4 + 5) ÷ 3 5 using PEMDAS.

Solution:

Step 1 – Parentheses : 2+6 × (4 + 5) ÷ 3 – 5 = 2 + 6 × 9 ÷ 3 – 5

Step 2 – Multiplication: 2 + 6 × 9 ÷ 3 – 5 = 2 + 54 ÷ 3 – 5

Step 3 – Division: 2 + 54 ÷ 3 – 5 = 2 + 18 – 5

Step 4 – Addition: 2 + 18 – 5 = 20 – 5

Step 5 – Subtraction: 20 5 = 15

Example 2: Solve 4 – 5 ÷ (8 – 3) × 2 + 5 using PEMDAS.

Solution:

Step 1 – Parentheses: 4 – 5 ÷ (8 – 3) × 2 + 5 = 4 – 5 ÷ 5 × 2 + 5

Step 2 – Division: 4 – 5 ÷ 5 × 2 + 5 = 4 – 1 × 2 + 5

Step 3 – Multiplication:  4 – 1 × 2 + 5 = 4 – 2 + 5

Step 4 – Subtraction: 4 – 2 + 5 = 2 + 5

Step 5 – Addition: 2 + 5 = 7

Example 3: Solve 100 ÷ (6 + 7 × 2) 5 using PEMDAS.

Solution:

Step 1 – Multiplication inside parentheses: 100 ÷ (6 + 7 × 2) – 5= 100 ÷ (6 + 14) – 5

Step 2 – Addition inside parentheses: 100 ÷ (6 + 14) – 5 = 100 ÷ 20 – 5

Step 3 – Division: 100 ÷ 20 – 5 = 5 – 5 Step 4 – Subtraction: 5 5 = 0

## Practice Problems

1

### Simplify 4+ (5 ×3² + 2) using PEMDAS.

48
49
50
51
CorrectIncorrect
Correct answer is: 51
4 + (5 × 3² + 2)
= 4 + (5 × 9 + 2
) = 4 + (45 + 2)
= 4 + 47
= 51
2

### Simplify 9 – 24 ÷ 8 × 2 + 3 using PEMDAS.

7
6
5
4
CorrectIncorrect
Correct answer is: 6
9 – 24 ÷ 8 × 2 + 3
= 9 – 3 × 2 + 3
= 9 – 6 + 3
= 3 + 3
= 6
3

### Simplify [(32 ÷ 4) + 3] × 2 using PEMDAS.

20
18
22
10
CorrectIncorrect
Correct answer is: 22
[(32 ÷ 4) + 3] × 2
= [8 + 3] × 2
= 11 × 2
= 22
4

### Simplify $(3 × 5² ÷ 5)$ – $(16 - 10)$ using PEMDAS.

15
9
3
0
CorrectIncorrect
Correct answer is: 9
$(3 × 5² ÷ 5)$ – $(16 – 10)$
= $(3 × 25 ÷ 5)$ – $(16 – 10)$
= $(75 ÷ 5)$ – $(6)$
= 15 – 6
= 9

## Frequently Asked Questions

The order of operations are the rules that tell us the sequence in which we should solve an expression with multiple operations.

The order is PEMDAS: Parentheses, Exponents, Multiplication, and Division (from left to right), Addition and Subtraction (from left to right).

Yes. You can use the phrase “Please Excuse My Dear Aunt Sally” to remember PEMDAS.

Yes. Both PEMDAS and BODMAS are acronyms for remembering the order of operations. They are different names for the same rule. What they call PEMDAS in the US is called BODMAS in the UK, Australia, India and various other countries.

NOTE – Related Readings: